Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C32×Q8 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | (C3⋊D3)⋊A4 |
| commutator, hypocenter | C32×Q8 | C3×C4 |
| Frattini, socle | C22×C3 | D3×A4 |
| Fitting, cosocle | C2×C32×Q8 | C2×C3 |
| prime radical | C2×Q8⋊H(3,𝔽3) | C2 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C22 | |
139 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3,2^3,3^2,2|a^2=c,d^2=h,e^2=h,f^a=f^2,g^a=g^2,d^b=e,e^b=deh,f^b=fg,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #13146328, #13146329, #13146333, … |
| maximal normal subgroup of | #13146328, #13146329, #13146333, … |
| maximal quotient of | #13136849, C2×#11720765 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 732);